R E S E A R C H
Open Access
Fuzzy stability of a cubic functional equation via
fixed point technique
Syed Abdul Mohiuddine
*and Abdullah Alotaibi
* Correspondence: [email protected] Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Abstract
The object of this article is to determine Hyers-Ulam-Rassias stability results
concerning the cubic functional equation in fuzzy normed space by using the fixed point method.
Keywords:Hyers-Ulam-Rassias stability, cubic functional equation, fuzzy normed space, fixed point
1 Introduction, definitions and notations
Fuzzy set theory is a powerful hand set for modeling uncertainty and vagueness in var-ious problems arising in the field of science and engineering. It has a large number of application, for instance, in the computer programming [1], engineering problems [2], statistical convergence [3-7], nonlinear operator [8], best approximation [9] etc. Parti-cularly, fuzzy differential equation is a strong topic with large application areas, for example, in population models [10], civil engineering [11] and so on.
By modifying own studies on fuzzy topological vector spaces, Katsaras [12] first introduced the notion of fuzzy seminorm and norm on a vector space and later on Felbin [13] gave the concept of a fuzzy normed space (for short, FNS) by applying the notion fuzzy distance of Kaleva and Seikala [14] on vector spaces. Further, Xiao and Zhu [15] improved a bit the Felbin’s definition of fuzzy norm of a linear operator between FNSs.
Stability problem of a functional equation was first posed by Ulam [16] which was answered by Hyers [17] under the assumption that the groups are Banach spaces. Rassias [18] and Gajda [19] considered the stability problem with unbounded Cauchy differences. The unified form of the results of Hyers, Rassias, and Gajda is as follows:
Let E and F be real normed spaces with F complete and let f: E® F be a mapping such that the following condition holds
fx+y−f(x)−fyF≤θxpE+ypE,
for all x, yÎE,θ ≥0and for some pÎ[0,∞) | {1}.Then there exists a unique addi-tive function C : E® F such that
f(x)−C(x)F≤ 2θ |2−2p|x
p E,
for all xÎE.
This stability phenomenon is called generalized Hyers-Ulam stability and has been extensively investigated for different functional equations. It is to be noted that almost all proofs used the idea imaginated by Hyers. Namely, the additive function C:E® F
is explicitly constructed, starting from the given function f, by the formulae (i)
C(x)= limn→∞21nf(2nx), if p < 1; and (ii) C(x)= limn→∞2nf
x
2n
, if p > 1. This
method is called a direct method. It is often used to construct a solution of a given functional equation and is seen to be a powerful tool for studying the stability of many functional equations. Since then several stability problems and its fuzzy version for var-ious functional equations have been investigated in [20-26]. Recently, Radu [27] pro-posed that fixed point alternative method is very useful for obtaining the solution of Ulam problem.
The stability problem for the cubic functional equation was proved by Jun and Kim [21] for mappings f : X® Y, where Xis a real normed space andYis a Banach space. In this article, we show that the existence of the limit C(x) and the estimation (i) and (ii) can be simply obtained from the alternative of fixed point.
In this section, we recall some notations and basic definitions used in this article. A fuzzy subset N ofX×ℝis called a fuzzy norm onX if the following conditions are satisfied for allx, yÎX andc Î ℝ;
(a)N(x, t) = 0 for all non-positivetÎℝ, (b)N(x, t) = 1 for alltÎ ℝ+ if and only ifx= 0,
(c)N(cx,t)=N
x,|tc|
for alltÎℝ+andc≠0,
(d)N(x + y, t + s)≥ min{N(x, t),N(y, s)} for alls, tÎℝ,
(e)N(x, t) is a non-decreasing function onℝ, and suptÎℝN(x,t) = 1.
The pair (X, N) will be referred to as afuzzy normed space. Example 1.1. Let (X, ||.||) be a normed linear space. Then
N(x,t)=
t
t+x ift>0,
0 ift≤0,
is a fuzzy norm on X.
Example 1.2. Let (X, ||.||) be a normed linear space. Then
N(x,t)=
1 ift>x, 0 ift≤ x,
is a fuzzy norm on X.
Let (X, N) be a fuzzy normed space. Then, a sequencex = (xk) is said to be fuzzy
convergent toLÎ X if limN(xk - L, t) = 1, for all t> 0. In this case, we write N-lim
xk=L.
Let (X, N) be an fuzzy normed space. Then, x = (xk) is said to be fuzzy Cauchy
sequenceif limN(xk+p- xk, t) = 1 for allt> 0 and p= 1, 2, ....
2 Fixed point technique for Hyers-Ulam stability
In this section, we deal with the stability problem via fixed point method in fuzzy norm space. Before proceeding further, we should recall the following results related to the concept of fixed point.
Theorem 2.1 (Banach’s Contraction principle). Let (X, d) be a complete generalized metric space and consider a mappingJ : X® Xbe a strictly contractive mapping, that is
dJx,Jy≤Ldx,y, ∀x,y∈X
for some (Lipschitz constant) L< 1. Then
(i) The mappingJ has one and only one fixed pointx* =J(x*); (ii) The fixed pointx* is globally attractive, that is
lim
n→∞J
nx=x∗,
for any starting point xÎX;
(iii) One has the following estimation inequalities for allxÎXand n≥0:
dJnx,x∗≤Lndx,x∗ (2:1:1)
dJnx,x∗≤ 1 1−Ld
Jnx,Jn+1x (2:1:2)
dx,x∗≤ 1
1−Ld(x,Jx). (2:1:3)
Theorem 2.2(The alternative of fixed point) [28]. Suppose we are given a complete generalized metric space (X, d) and a strictly contractive mappingJ : X ® X, with Lipschitz constantL. Then, for each given elementxÎX, either
dJnx,Jn+1x= +∞, ∀n≥0 (2:2:1)
or
d(Jnx, Jn+1x)<+∞ ∀n≥n◦ (2:2:2)
for some natural numbern0. Moreover, if the second alternative holds then
(i) The sequence (Jnx) is convergent to a fixed pointy* ofJ;
(ii)y* is the unique fixed point ofJin the setY ={y∈X, d(Jn◦x, y)<+∞} (iii)dy,y∗≤ 1
1−Ld
y,Jy,y∈Y.
We are now ready to obtain our main results. The functional equation
is called the cubic functional equation, since the functionf(x) =cx3 is its solution. Every solution of the cubic functional equation is said to be acubic mapping.
Letbe a function from X×Xto Z. A mappingf: X® Yis said to be -approxi-mately cubic function if
Nf2x+y+f2x−y−2fx+y−2fx−y−12f(x),t≥Nϕx,y,t (2:0:2)
for all x, yÎX and t> 0.
Using the fixed point alternative, we can prove the stability of Hyers-Ulam-Rassias type theorem in FNS. First, we prove the following lemma which will be used in our main result.
Lemma 2.1. Let (Z, N’) be a fuzzy normed space and:X® Zbe a function. LetE
= {g:X®Y; g(0) = 0} and define
dM
g,h= infa∈R+:Ng(x)−h(x),at≥N(ϕ (x, 0),t) for allx∈Xandt>0 ,
for all hÎ E. ThendMis a complete generalized metric onE.
Proof. Letg, h, kÎ E, dM(g, h) <ξ1anddM(h, k) <ξ2. Then
Ng(x)−h(x),ξ1t≥N(ϕ (x),t) andN(h(x)−k(x),ξ2t)≥N(ϕ (x),t),
for all xÎXand t> 0. Thus
Ng(x)−k(x),(ξ1+ξ2)t
≥minNg(x)−h(x),ξ1t
,N(h(x)−k(x),ξ2t) ≥N(ϕ (x),t),
for each x Î X and t> 0. By definitiondM(h, k) <ξ1 + ξ2. This proves the triangle inequality for dM. Rest of the proof can be done on the same lines as in (see [[29],
Lemma 2.1]).
Theorem 2.3. LetX be a linear space and (Z, N’) be a FNS. Suppose that a function
: X×X ®Z satisfying(2x, 2y) = a(x, y) for allx, y ÎX anda≠0. Suppose that (Y, N) be a fuzzy Banach space andf: X ®Ybe a-approximately cubic function. If for some 0 <a< 8
N(ϕ(2x, 0),t)≥N(αϕ(x, 0),t), (2:3:1) and
lim
n→∞N
(ϕ(2nx, 2ny), 8nt) = 1,
for allx, yÎ Xand t> 0. Then there exists a unique cubic mappingC : X® Ysuch that
N(C(x)−f(x),t)≥N(ϕ(x, 0), 2(8−α)t),
for all xÎXand all t> 0.
Proof. Puty= 0 in (2.0.2). Then for allxÎX andt> 0
N
f(2x) 8 −f(x),
t 16
≥N(ϕ(x, 0),t).
Consider the setE = {g:X ®Y, g(0) = 0} together with the mappingdMdefined on
E × E by
It is known thatdM(g, h) complete generalized metric space by Lemma 2.1. Now, we
define the linear mapping J:E® Esuch that
Jg(x) = 1 8g(2x).
It is easy to see that J is a strictly contractive self-mapping ofE with the Lipschitz
constantα
Replacingtby 2(8 -a)tin the above equation, we obtain
N(C(x)−f(x),t)≥N(ϕ(x, 0), 2(8−α)t),
for all xÎXand t> 0. Let x, yÎ X. Then
N(C(2x+y) +C(2x−y)−2C(x+y)−2C(x−y)−12C(x),t)≥N(ϕ(x,y),t).
Replacingxand yby 2nxand 2ny, respectively, we obtain
N
C(2n(2x+y)) 8n +
C(2n(2x−y)) 8n −
2C(2n(x+y)) 8n −
2C(2n(x−y)) 8n −
12C(2nx) 8n ,t
≥N(ϕ(2nx, 2ny), 8nt)
for all x, yÎ Xand allt> 0. Since
lim
n→∞N
(ϕ(2nx, 2ny), 8nt) = 1,
we conclude that Cfulfills (2.0.1).
The uniqueness of Cfollows from the fact thatCis the unique fixed point ofJwith the following property that there existsuÎ(0,∞) such that
N(C(x)−f(x),ut)≥N(ϕ(x, 0),t),
for all xÎXand t> 0.
This completes the proof of the theorem.
By a modification in the proof of Theorem 2.3, one can prove the following:
Theorem 2.4. LetX be a linear space and (Z, N’) be a FNS. Suppose that a function
:X×X®Zsatisfying
ϕx
2, y 2
= 1
αϕ(x,y)
for all x, yÎXand a≠0. Suppose that (Y, N) be a fuzzy Banach space andf:X ¬Y
be a-approximately cubic function. If for somea> 8
N(ϕ(x/2, 0),t)≥N(ϕ(x, 0),αt),
and
lim
n→∞N
(8−nϕ(2−nx, 2−ny),t) = 1
for allx, yÎ Xandt> 0. Then there exists a unique cubic mappingC:X ®Ysuch that
N(C(x)−f(x),t)≥N(ϕ(x, 0), 2(α−8)t),
for all xÎXand all t> 0.
The proof of the above theorem is similar to the proof of Theorem 2.3, hence omitted.
3 Conclusion
by other authors for fuzzy setting. We established Hyers-Ulam-Rassias stability of a cubic functional equation in fuzzy normed spaces by using fixed point alternative theorem.
Author’information
Address of both the authors: Department of Mathematics, Faculty of Science, King Abdu-laziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. Email: [email protected] (S.A. Mohiuddine); [email protected] (A. Alotaibi).
Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments.
Authors’contributions
Both the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 20 January 2012 Accepted: 17 April 2012 Published: 17 April 2012
References
1. Giles, R: A computer program for fuzzy reasoning. Fuzzy Sets Syst.4, 221–234 (1980)
2. Hanss, M: Applied fuzzy arithmetic: an introduction with engineering applications. Springer-Verlag, Berlin (2005) 3. Mohiuddine, SA, Danish Lohani, QM: On generalized statistical convergence in intuitionistic fuzzy normed space. Chaos
Solitons Fract.42, 1731–1737 (2009)
4. Mursaleen, M, Mohiuddine, SA: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Solitons Fract.41, 2414–2421 (2009)
5. Mursaleen, M, Mohiuddine, SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J Comput Appl Math.233(2):142–149 (2009)
6. Mursaleen, M, Mohiuddine, SA, Edely, OHH: On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces. Comput Math Appl.59, 603–611 (2010)
7. Mohiuddine, SA,Şevli, H, Cancan, M: Statistical convergence in fuzzy 2-normed space. J Computational Analy Appl.
12(4):787–798 (2010)
8. Mursaleen, M, Mohiuddine, SA: Nonlinear operators between intuitionistic fuzzy normed spaces and Fr’echet differentiation. Chaos Solitons Fract.42, 1010–1015 (2009)
9. Mohiuddine, SA: Some new results on approximation in fuzzy 2-normed spaces. Math Comput Model.53, 574–580 (2011)
10. Guo, M, Li, R: Impulsive functional differential inclusions and fuzzy population models. Fuzzy Sets Syst.138, 601–615 (2003)
11. Oberguggenberger, M, Pittschmann, S: Differential equations with fuzzy parameters. Math Mod Syst.5, 181–202 (1999) 12. Katsaras, AK: Fuzzy topological vector spaces. Fuzzy Sets Syst.12, 143–154 (1984)
13. Felbin, C: Finite dimensional fuzzy normed linear spaces. Fuzzy Sets Syst.48, 239–248 (1992) 14. Kaleva, O, Seikala, S: On fuzzy metric spaces. Fuzzy Sets Syst.12, 215–229 (1984)
15. Xiao, JZ, Zhu, XH: Fuzzy normed spaces of operators and its completeness. Fuzzy Sets Syst.133(3):135–146 (2003) 16. Ulam, SM: A Collection of the Mathematical Problems. Interscience Publ. New York (1960)
17. Hyers, DH: On the stability of the linear functional equation. Proc Natl Acad Sci USA.27, 222–224 (1941) 18. Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc.72, 297–300 (1978) 19. Gajda, Z: On stability of additive mappings. Int J Math Math Sci.14, 431–434 (1991)
20. Alotaibi, A, Mohiuddine, SA: On the stability of a cubic functional equation in random 2-normed spaces. Adv Diff Equ.
2012, 39 (2012)
21. Jun, KW, Kim, HM: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J Math Anal Appl.274, 867–878 (2002)
22. Jung, SM, Kim, TS: A fixed point approach to the stability of the cubic functional equation. Bol Soc Mat Mexicana.
12(1):51–57 (2006)
23. Mohiuddine, SA: Stability of Jensen functional equation in intuitionistic fuzzy normed space. Chaos Solitons Fract.42, 2989–2996 (2009)
24. Mohiuddine, SA,Şevli, H: Stability of Pexiderized quadratic functional equation in intuitionistic fuzzy normed space. J Comput Appl Math.235, 2137–2146 (2011)
25. Mursaleen, M, Mohiuddine, SA: On stability of a cubic functional equation in intuition-istic fuzzy normed spaces. Chaos Solitons Fract.42, 2997–3005 (2009)
26. Mirmostafaee, AK: A fixed point approach to almost quartic mappings in quasi fuzzy normed spaces. Fuzzy Sets Syst.
160, 1653–1662 (2009)
27. Radu, V: The fixed point alternative and the stability of functional equations. Sem Fixed Point Theory.4(1):91–96 (2003) 28. Diaz, JB, Margolis, B: A fixed point theorem of the alternative for contractions on generalized complete metric space.
29. Miheţ, D, Radu, V: On the stability of the additive Cauchy functional equation in random normed spaces. J Math Anal Appl.343, 567–572 (2008)
doi:10.1186/1687-1847-2012-48
Cite this article as:Mohiuddine and Alotaibi:Fuzzy stability of a cubic functional equation via fixed point technique.Advances in Difference Equations20122012:48.
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